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Produced with R version 3.2.1 and pomp version 0.74.2.

Objectives

  1. To convey and understanding of the nature of the problem of likelihood computation for POMP models
  2. To explain the simplest particle filter algorithm
  3. To give students experience visualizing and exploring likelihood surfaces using the particle filter for computation of the likelihood
  4. Understand the basic techniques of likelihood-based inference
  5. Describe how to apply these techniques in situations where the likelihood cannot be written down explicitly but can be evaluated and maximized via Monte Carlo methods.
  6. Gain some experience at carrying out likelihood-based inferences for dynamic models using simulation-based statistical methodology in the R package ‘pomp’.

Theory of the particle filter

The likelihood function

  • The basis for modern frequentist, Bayesian, and information-theoretic inference.
  • The function itself is a representation of the what the data have to say about the parameters.
  • A good general reference on likelihood is Pawitan (2001).

Definition of the likelihood function

Data are a sequence of \(N\) observations, denoted \(y_{1:N}^*\). A statistical model is a density function \(f(y_{1:N};\theta)\) which defines a probability distribution for each value of a parameter vector \(\theta\). Statistical inference involves deciding for which (if any) values of \(\theta\) it is reasonable to model \(y_{1:N}^*\) as a random draw from \(f(y_{1:N};\theta)\).

The likelihood function is the density function evaluated at the data. It is usually convenient to work with the log likelihood function, \[{\ell}(\theta)=\log f(y_{1:N}^*;\theta)\]

Modeling using discrete and continuous distributions

Recall that the probability distribution \(f(y_{1:N};\theta)\) defines a random variable \(Y_{1:N}\) for which probabilities can be computed as integrals of \(f(y_{1:N};\theta)\). Specifically, for any event \(E\) describing a set of possible outcomes of \(Y_{1:N}\), \[P[\mbox{$Y_{1:N}$ is in $E$}] = \int_E f(y_{1:N}^*;\theta)\, dy_{1:N}.\] If the model corresponds to a discrete distribution, then the integral is replaced by a sum and the probability density function is called a probability mass function. The definition of the likelihood function remains unchanged. We will use the notation of continuous random variables, but all the methods apply also to discrete models.

Indirect specification of the statistical model via a simulation procedure

  • For simple statistical models, we may describe the model by explicitly writing the density function \(f(y_{1:N};\theta)\). One may then ask how to simulate a random variable \(Y_{1:N}\sim f(y_{1:N};\theta)\).

  • For many dynamic models it is convenient to define the model via a procedure to simulate the random variable \(Y_{1:N}\). This implicitly defines the corresponding density \(f(y_{1:N};\theta)\). For a complicated simulation procedure, it may be difficult or impossible to write down \(f(y_{1:N};\theta)\) exactly.

  • It is important for us to bear in mind that the likelihood function exists even when we don’t know what it is! We can still talk about the likelihood function, and develop numerical methods that take advantage of its statistical properties.

Likelihood for POMP models

state_space_diagram2

POMP model schematic, showing dependence among model variables. State variables, \(x\), at time \(t\) depend only on state variables at the previous timestep. Measurements, \(y\), at time \(t\) depend only on the state at that time.

POMP model notation

  • Write \(X_n=X(t_n)\) and \(X_{0:N}=(X_0,\dots,X_N)\).

  • Let \(Y_n\) be a random variable modeling the observation at time \(t_n\).

  • The one-step transition density, \(f_{X_n|X_{n-1}}(x_n|x_{n-1};\theta)\), together with the measurement density, \(f_{Y_n|X_n}(y_n|x_n;\theta)\) and the initial density, \(f_{X_0}(x_0;\theta)\), specify the entire joint density via \[f_{X_{0:N},Y_{1:N}}(x_{0:N},y_{1:N};\theta) = f_{X_0}(x_0;\theta)\,\prod_{n=1}^N\!f_{X_n | X_{n-1}}(x_n|x_{n-1};\theta)\,f_{Y_n|X_n}(y_n|x_n;\theta).\]

  • The marginal density for sequence of measurements, \(Y_{1:N}\), evaluated at the data, \(y_{1:N}^*\), is \[{\mathscr{L}}(\theta) = f_{Y_{1:N}}(y^*_{1:N};\theta)=\int\!f_{X_{0:N},Y_{1:N}}(x_{0:N},y^*_{1:N};\theta)\, dx_{0:N}.\]

Special case: deterministic unobserved state process

Lets’ begin with a special case. Suppose that the unobserved state process is deterministic. That is, \(X_{n}=x_n(\theta)\) is a known function of \(\theta\) for each \(n\). What is the likelihood?

Since the probability of the observation, \(Y_n\), depends only on \(X_n\) and \(\theta\), and since, in particular \(Y_{m}\) and \(Y_{n}\) are independent given \(X_{m}\) and \(X_{n}\), we have \[{\mathscr{L}}(\theta) = \prod_{n} f_{Y_n|X_n}(y_n^*;x_n(\theta),\theta)\] or \[\ell(\theta) = \log{\mathscr{L}}(\theta) = \sum_{n} \log f_{Y_n|X_n}(y_n^*;x_n(\theta),\theta).\]

Likelihood for stochastic models

We can imagine using Monte Carlo integration for computing the likelihood of a state space model, \[{\mathscr{L}}(\theta)={\mathbb{P}\left[{y_{1:T}|\theta}\right]}=\sum_{x_1}\cdots\sum_{x_T}\!\prod_{t=1}^{T}\!{\mathbb{P}\left[{y_t|x_t,\theta}\right]}\,{\mathbb{P}\left[{x_t|x_{t-1},\theta}\right]}.\] Specifically, if we have some probabilistic means of proposing trajectories for the unobserved state process, then we could just generate a large number of these and approximate \({\mathscr{L}}(\theta)\) by its Monte Carlo estimate. Specifically, we generate \(N\) trajectories of length \(T\), \(x_{t,k}\), \(k=1\,\dots,N\), \(t=1,\dots,T\). Let \(w_k\) denote the probability of proposing trajectory \(k\). For each trajectory, we compute the likelihood of that trajectory \[{\mathscr{L}}{_k}(\theta)=\prod_{t=1}^{T} {\mathbb{P}\left[{y_t|x_{t,k},\theta}\right]}\,{\mathbb{P}\left[{x_{t,k}|x_{t-1,k},\theta}\right]}\] Then by the Monte Carlo theorem, we have \[{\mathscr{L}}(\theta) \approx \frac{1}{N}\,\sum_{k=1}^{N}\!\frac{{\mathscr{L}}_k(\theta)}{w_k}.\]

How shall we choose our trajectories? One idea would be to choose them so as to simplify the computation. If we choose them such that \[w_k=\prod_{t=1}^{T} {\mathbb{P}\left[{x_{t,k}|x_{t-1,k},\theta}\right]},\] then we have \[{\mathscr{L}}(\theta) \approx \frac{1}{N}\,\sum_{k=1}^{N} \frac{{\mathscr{L}}_k(\theta)}{w_k} = \frac{1}{N}\,\sum_{k=1}^{N}\!\frac{\prod_{t=1}^{T} {\mathbb{P}\left[{y_t|x_{t,k},\theta}\right]}\,{\mathbb{P}\left[{x_{t,k}|x_{t-1,k},\theta}\right]}}{\prod_{t=1}^{T} {\mathbb{P}\left[{x_{t,k}|x_{t-1,k},\theta}\right]}} = \frac{1}{N}\,\sum_{k=1}^{N}\!\prod_{t=1}^{T} {\mathbb{P}\left[{y_t|x_{t,k},\theta}\right]}\]

This implies that if we generate trajectories by simulation, all we have to do is compute the likelihood of the data with given each trajectory and average.

Let’s go back to the boarding school influenza outbreak to see what this would look like. Let’s reconstruct the toy SIR model we were working with. Download a script containing the R commands below.

base_url <- "http://kingaa.github.io/sbied/"
url <- paste0(base_url,"data/bsflu_data.txt")
bsflu <- read.table(url)

sir_step <- Csnippet("
  double dN_SI = rbinom(S,1-exp(-Beta*I/N*dt));
  double dN_IR = rbinom(I,1-exp(-gamma*dt));
  S -= dN_SI;
  I += dN_SI - dN_IR;
  R += dN_IR;
  H += dN_IR;
")

sir_init <- Csnippet("
  S = nearbyint(N)-1;
  I = 1;
  R = 0;
  H = 0;
")

dmeas <- Csnippet("lik = dbinom(B,H,rho,give_log);")
rmeas <- Csnippet("B = rbinom(H,rho);")

pomp(bsflu,times="day",t0=0,
     rprocess=euler.sim(sir_step,delta.t=1/5),
     initializer=sir_init,rmeasure=rmeas,dmeasure=dmeas,
     zeronames="H",statenames=c("H","S","I","R"),
     paramnames=c("Beta","gamma","rho","N")) -> sir

Let’s generate a bunch of simulated trajectories at some particular point in parameter space.

simulate(sir,params=c(Beta=2,gamma=1,rho=0.5,N=2600),
         nsim=10000,states=TRUE) -> x
matplot(time(sir),t(x["H",1:50,]),type='l',lty=1,
        xlab="time",ylab="H",bty='l',col='blue')
lines(time(sir),obs(sir,"B"),lwd=2,col='black')

We can use the function dmeasure to evaluate the log likelihood of the data given the states, the model, and the parameters:

ell <- dmeasure(sir,y=obs(sir),x=x,times=time(sir),log=TRUE,
                params=c(Beta=2,gamma=1,rho=0.5,N=2600))
dim(ell)
## [1] 10000    14

According to the equation above, we should sum up the log likelihoods across time:

ell <- apply(ell,1,sum); summary(exp(ell)); logmeanexp(ell,se=TRUE)
##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
##  0.000e+00  0.000e+00  0.000e+00 7.760e-108  0.000e+00 7.754e-104
##                    se 
## -246.63090   99.99759

The likelihood appears to be very low, but the error in the estimate is very high and therefore the estimated likelihood is very imprecise. We are going to need very many simulations to get an estimate of the likelihood sufficiently precise to be of any use in parameter estimation or model selection.

What’s the problem? Essentially, far too many of the trajectories don’t pass near the data. Moreover, once a trajectory diverges from the data, it almost never comes back. This is a consequence of the fact that we are proposing trajectories in a way that is completely unconditional on the data. The problem will only get worse with longer data sets!

The particle filter

We can arrive at a more efficient algorithm by factorizing the likelihood in a different way. \[{\mathscr{L}}(\theta)={\mathbb{P}\left[{y^*_{1:T}|\theta}\right]} =\prod_{t}\,{\mathbb{P}\left[{y^*_t|y^*_{1:t-1},\theta}\right]} =\prod_{t}\,\sum_{x_{t}}\,{\mathbb{P}\left[{y^*_t|x_t,\theta}\right]}\,{\mathbb{P}\left[{x_t|y^*_{1:t-1},\theta}\right]}\] Now the Markov property gives us that \[{\mathbb{P}\left[{x_t|y^*_{1:t-1},\theta}\right]} = \sum_{x_{t-1}}\,{\mathbb{P}\left[{x_t|x_{t-1},\theta}\right]}\,{\mathbb{P}\left[{x_{t-1}|y^*_{1:t-1},\theta}\right]}\] and Bayes’ theorem tells us that \[{\mathbb{P}\left[{x_t|y^*_{1:t},\theta}\right]} = {\mathbb{P}\left[{x_t|y^*_t,y^*_{1:t-1},\theta}\right]} =\frac{{\mathbb{P}\left[{y^*_{t}|x_{t},\theta}\right]}\,{\mathbb{P}\left[{x_{t}|y^*_{1:t-1},\theta}\right]}}{\sum_{x_{t}}\,{\mathbb{P}\left[{y^*_{t}|x_{t},\theta}\right]}\,{\mathbb{P}\left[{x_{t}|y^*_{1:t-1},\theta}\right]}}.\]

This suggests that we keep track of two key distributions. We’ll refer to the distribution of \(x_t | y^*_{1:t-1}\) as the prediction distribution at time \(t\) and the distribution of \(x_{t} | y^*_{1:t}\) as the filtering distribution at time \(t\).

Let’s use Monte Carlo techniques to estimate the sums. Suppose \(x_{t-1,k}^{F}\), \(k=1,\dots,N\) is a set of points drawn from the filtering distribution at time \(t-1\). We obtain a sample \(x_{t,k}^{P}\) of points drawn from the prediction distribution at time \(t\) by simply simulating the process model: \[x_{t,k}^{P} \sim \mathrm{process}(x_{t-1,k}^{F},\theta), \qquad k=1,\dots,N.\] Having obtained \(x_{t,k}^{P}\), we obtain a sample of points from the filtering distribution at time \(t\) by resampling from \(x_{t,k}^{P}\) with weights \({\mathbb{P}\left[{y^*_{t}|x_{t},\theta}\right]}\). The Monte Carlo theorem tells us, too, that the conditional likelihood \[{\mathscr{L}}_t(\theta) = {\mathbb{P}\left[{y^*_t|y^*_{1:t-1},\theta}\right]} = \sum_{x_{t}}\,{\mathbb{P}\left[{y^*_{t}|x_{t},\theta}\right]}\,{\mathbb{P}\left[{x_{t}|y^*_{1:t-1},\theta}\right]} \approx \frac{1}{N}\,\sum_k\,{\mathbb{P}\left[{y^*_{t}|x_{t,k}^{P},\theta}\right]}.\] We can iterate this procedure through the data, one step at a time, alternately simulating and resampling, until we reach \(t=T\). The full log likelihood is then approximately \[{\ell}(\theta) = \log{{\mathscr{L}}(\theta)} = \sum_t \log{{\mathscr{L}}_t(\theta)}.\]

This is known as the sequential Monte Carlo algorithm or the particle filter. Key references include Kitagawa (1987), Arulampalam et al. (2002) and the book by Doucet et al. (2001).

Sequential Monte Carlo in pomp

Here, we’ll get some practical experience with the particle filter, and the likelihood function, in the context of our influenza-outbreak case study.

sims <- simulate(sir,params=c(Beta=2,gamma=1,rho=0.8,N=2600),nsim=20,
                 as.data.frame=TRUE,include.data=TRUE)

ggplot(sims,mapping=aes(x=time,y=B,group=sim,color=sim=="data"))+
  geom_line()+guides(color=FALSE)

In pomp, the basic particle filter is implemented in the command pfilter. We must choose the number of particles to use by setting the Np argument.

pf <- pfilter(sir,Np=5000,params=c(Beta=2,gamma=1,rho=0.8,N=2600))
logLik(pf)
## [1] -87.17785

We can run a few particle filters to get an estimate of the Monte Carlo variability:

pf <- replicate(10,pfilter(sir,Np=5000,params=c(Beta=2,gamma=1,rho=0.8,N=2600)))
ll <- sapply(pf,logLik); ll
##  [1] -75.66538 -76.96993 -75.89798 -87.27089 -80.75397 -88.18815 -82.97545
##  [8] -71.30925 -70.89917 -74.57561
logmeanexp(ll,se=TRUE)
##                    se 
## -72.667289   2.094562

Note that we’re careful here to counteract Jensen’s inequality. The particle filter gives us an unbiased estimate of the likelihood, not of the log-likelihood.

The graph of the likelihood function: The likelihood surface

To get an idea of what the likelihood surface looks like in the neighborhood of the default parameter set supplied by sir, we can construct some likelihood slices. We’ll make slices in the \(\beta\) and \(\gamma\) directions. Both slices will pass through the default parameter set.

sliceDesign(
  c(Beta=2,gamma=1,rho=0.8,N=2600),
  Beta=rep(seq(from=0.5,to=4,length=40),each=3),
  gamma=rep(seq(from=0.5,to=2,length=40),each=3)) -> p

require(foreach)
require(doMC)
registerDoMC(cores=5)        ## number of cores on this machine

set.seed(998468235L,kind="L'Ecuyer")
mcopts <- list(preschedule=FALSE,set.seed=TRUE)

foreach (theta=iter(p,"row"),.combine=rbind,
         .inorder=FALSE,.options.multicore=mcopts) %dopar% 
 {
   pfilter(sir,params=unlist(theta),Np=5000) -> pf
   theta$loglik <- logLik(pf)
   theta
 } -> p

Note that we’ve used the foreach package with the multicore backend (doMC) to parallelize these computations. To ensure that we have high-quality random numbers in each parallel R session, we use a parallel random number generator (kind="L'Ecuyer", .options.multicore=list(set.seed=TRUE)).

foreach (v=c("Beta","gamma")) %do% 
{
  x <- subset(p,slice==v)
  plot(x[[v]],x$loglik,xlab=v,ylab="loglik")
}

Exercise: Likelihood slices

Add likelihood slices along the \(\rho\) direction.

Slices offer a very limited perspective on the geometry of the likelihood surface. With just two parameters, we can evaluate the likelihood at a grid of points and visualize the surface directly.

expand.grid(Beta=seq(from=1,to=4,length=50),
            gamma=seq(from=0.7,to=3,length=50),
            rho=0.8,
            N=2600) -> p

foreach (theta=iter(p,"row"),.combine=rbind,
         .inorder=FALSE,.options.multicore=mcopts) %dopar% 
 {
   pfilter(sir,params=unlist(theta),Np=5000) -> pf
   theta$loglik <- logLik(pf)
   theta
 } -> p
pp <- mutate(p,loglik=ifelse(loglik>max(loglik)-100,loglik,NA))
ggplot(data=pp,mapping=aes(x=Beta,y=gamma,z=loglik,fill=loglik))+
  geom_tile(color=NA)+
  geom_contour(color='black',binwidth=3)+
  scale_fill_gradient()+
  labs(x=expression(beta),y=expression(gamma))

Exercise: 2D likelihood slice

Compute a slice of the likelihood in the \(\beta\)-\(N\) plane.

Maximizing the likelihood

Call the whole parameter space \(\Theta\). Let \(\Theta^*\) be a subset of \(\Theta\), constraining parameters to describe scientific hypotheses of interest. For example, in a disease transmission model, \(\Theta^*\) could assert that the probability of a case being reported is \(\rho=0.8\).

Exercise: AIC as a formal statistical test

Determine the size of AIC as a hypothesis test for nested hypotheses with \(d=1\) in a regular parametric situation.

Confidence intervals for parameters: Profile likelihood

The likelihood ratio test with \(d=1\) gives a good way to construct confidence intervals. Suppose we are interested in a specific parameter, \(\phi\), and we want to consider whether the data support a possibility that \(\phi=\phi^*\) in the absence of assumptions on the other parameters. Then, we can take \(\Theta^*\) to be the subset of \(\Theta\) satisfying \(\phi=\phi^*\). Using the chi-square approximation to the likelihood ratio statistic, a 95% confidence interval for \(\phi\) consists of all the values \(\phi^*\) for which \[\ell_\mathrm{max}-\ell_\mathrm{max}^* < 1.92.\]

A way to vizualize the information about a specific parameter \(\phi\) is via the profile likelihood function, defined as \[\ell_\mathrm{profile}(\phi^*) = \max\{\mbox{$\ell(\theta)$ subject to the constraint $\phi=\phi^*$}\}.\] We then plot \(\ell_\mathrm{profile}(\phi)\) against \(\phi\).

Point estimates for parameters: The maximum likelihood estimate (MLE)

We define maximum likelihood estimates (MLEs) \(\hat\theta\) and \(\hat\theta^*\) such that \[\ell(\hat\theta)=\ell_\mathrm{max},\quad \ell(\hat\theta^*)=\ell_\mathrm{max}^*.\]

Biological interpretation of parameter estimates

When we write down a mechanistic model for an epidemiological system, we have some idea of what we intend parameters to mean; a reporting rate, a contact rate between individuals, an immigration rate, a duration of immunity, etc.

Maximizing the likelihood using the particle filter

Clearly, the default parameter set is not particularly close to the MLE. One way to find the MLE is to try optimizing the estimated likelihood directly. There are many optimization algorithms to choose from, and many implemented in R.

Three issues arise immediately.

  1. The particle filter gives us a stochastic estimate of the likelihood. We can reduce this variability by making Np larger, but we cannot make it go away. If we use a deterministic optimizer (i.e., one that assumes the objective function is evaluated deterministically), then we must control this variability somehow. For example, we can fix the seed of the pseudo-random number generator. A side effect will be that the objective function becomes jagged, marked by many small local knolls and pits. Alternatively, we can use a stochastic optimization algorithm, with which we will be only be able to obtain estimates of our MLE. This is the trade-off between a noisy and a rough objective function.
  2. Because the particle filter gives us just an estimate of the likelihood and no information about the derivative, we must choose an algorithm that is “derivative-free”. There are many such, but we can expect less efficiency than would be possible with derivative information. Note that finite differencing is not an especially promising way of constructing derivatives. The price would be a \(n\)-fold increase in cpu time, where \(n\) is the dimension of the parameter space. Also, since the likelihood is only estimated, we would expect the derivative estimates to be noisy.
  3. Finally, the parameters are constrained to be positive, and \(\rho < 1\). We must therefore select an optimizer that can solve this constrained maximization problem, or figure out some of way of turning it into an unconstrained maximization problem. For the latter purpose, one can transform the parameters onto a scale on which there are no constraints.

Here, let’s opt for deterministic optimization of a rough function. We’ll try using optim’s default method: Nelder-Mead, fixing the random-number generator seed to make the likelihood calculation deterministic. Since Nelder-Mead is an unconstrained optimizer, we must transform the parameters. The following Csnippets encode an appropriate transformation and its inverse, and introduce them into the pomp object.

toEst <- Csnippet("
 TBeta = log(Beta);
 Tgamma = log(gamma);
 Trho = logit(rho);
")

fromEst <- Csnippet("
 TBeta = exp(Beta);
 Tgamma = exp(gamma);
 Trho = expit(rho);
")

pomp(sir,toEstimationScale=toEst,
     fromEstimationScale=fromEst,
     paramnames=c("Beta","gamma","rho")) -> sir

Let’s fix a reference point in parameter space and insert these parameters into the pomp object:

coef(sir) <- c(Beta=2,gamma=1,rho=0.8,N=2600)

The following constructs a function returning the negative log likelihood of the data at a given point in parameter space.

neg.ll <- function (par, est) {
  ## parameters to be estimated are named in 'est'
  allpars <- coef(sir,transform=TRUE)
  allpars[est] <- par
  try(
    freeze(
      pfilter(sir,params=partrans(sir,allpars,dir="fromEst"),
              Np=1000),
      seed=5859684
    )
  ) -> pf
  if (inherits(pf,"try-error")) {
    1e10                 ## a big, bad number
    } else {
      -logLik(pf)
    }
}

Now we call optim to minimize this function:

## use Nelder-Mead with fixed RNG seed
fit <- optim(
  par=c(log(1), log(2), log(0.8)),
  est=c("gamma","Beta","rho"),
  fn=neg.ll,
  method="Nelder-Mead",
  control=list(maxit=400,trace=0)
)

mle <- sir
coef(mle,c("gamma","Beta","rho"),transform=TRUE) <- fit$par
coef(mle,c("gamma","Beta","rho")) ## point estimate
##     gamma      Beta       rho 
## 0.4989446 1.8519686 0.6079366
fit$val
## [1] 68.84983
simulate(mle,nsim=8,as.data.frame=TRUE,include=TRUE) -> sims

lls <- replicate(n=5,logLik(pfilter(mle,Np=5000)))
ll <- logmeanexp(lls,se=TRUE); ll
##                      se 
## -72.7318951   0.4081526

Some simulations at these parameters are shown next:

ggplot(data=sims,mapping=aes(x=time,y=B,group=sim,color=sim=="data"))+
  geom_line()

Exercise: More slices

Construct likelihood slices through the MLE we just found.

Exercise: Visualizing the likelihood surface

Evaluate the likelihood at points on a grid lying in a 2D slice through the MLE we found above. Each group should choose a different slice. Afterward, we’ll compare results across groups.

Exercise: Global maximization

The search of parameter space we conducted above was local. It is possible that we found a local maximum, but that other maxima exist with higher likelihoods. Conduct a more thorough search by initializing the Nelder-Mead starting points across a wider region of parameter space. Do you find any other local maxima?

References

Arulampalam, M. S., S. Maskell, N. Gordon, and T. Clapp. 2002. A tutorial on particle filters for online nonlinear, non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing 50:174–188.

Doucet, A., N. de Freitas, and N. Gordon, editors. 2001. Sequential Monte Carlo Methods in Practice. Springer-Verlag, New York.

Kitagawa, G. 1987. Non-Gaussian state-space modeling of nonstationary time series. Journal of the American Statistical Association 82:1032–1041.

Pawitan, Y. 2001. In All Likelihood: Statistical Modelling and Inference Using Likelihood. Clarendon Press, Oxford.